Before studying, in Chapters 19 and 20, some nonlinear wave equations occuring in mechanics, we consider in this chapter several linear wave equations arising in mechanics and study some fundamental aspects of the corresponding vibration phenomena.

In Section 18.1, we start by recalling the fundamental wave equations that appeared in the previous chapters: the equations of linear acoustics that appeared in Chapter 8 in the context of fluid mechanics and the Navier equation that appeared in Chapter 13 in the context of linear elasticity. We also specialize these equations to specific phenomena such as sound pipes and vibrating cords and membranes.

In Section 18.2, we show how to solve the one-dimensional wave equation considered in the whole space ℝ. Then, in Section 18.3, we are interested in bounded intervals, leading us to introduce the normal (self-vibration) modes, which also depend on the boundary conditions; some typical examples of boundary conditions are considered and the corresponding eigenmodes made explicit. In Section 18.4, we show how to solve the wave equation in a general bounded domain of ℝ3 by using the corresponding eigenmodes again; however, in this case, the solution is not complete because we cannot compute the eigenmodes, in general. Finally, in the last section of this chapter, Section 18.5, we give some indications on other important vibration phenomena such as superposition of waves, beats, and wave packets.

Returning to the equations of linear acoustics and of linear elasticity

Returning to the wave equations of linear acoustics

The equations of linear acoustics have been introduced in Chapter 8, Section 8.4.

Introduction

The problem of fluid dynamic behavior in spinning tanks is encountered in the study of stability and control of rockets, space vehicles, liquid-cooled gas turbines, centrifuges, and oceans. Some spacecraft vehicles are designed to spin in order to gain gyroscopic stiffness during the transfer from low earth orbit. This spin helps to control liquid propellant location in its container. Another class of problems deals with the dynamics and stability of rotating rigid bodies as applied to the evolution of celestial bodies and astronavigation control. Stabilization is achieved when the spacecraft spins about its axis of minimum moment of inertia. A satellite that spins about its axis of minimum moment of inertia may experience instability in the presence of energy dissipation. This is similar to a spinning top on a rough surface and as a result of friction the top's nutation angle increases as it seeks to conserve angular momentum.

The theory of rotating fluids is well documented in Lamb (1945), Morgan (1951), Howard (1963), and Greenspan (1968). Usually, the fluid motion is characterized by certain types of boundary layers and dissipative behavior in addition to complicated viscous layers controlled processes for spin-up and spin-down (Wilde and Vanyo, 1993). Lighthill (1966) presented a survey on the dynamics of rotating fluid. The results always have to be distinguished by the magnitude of the ratio of spin speed and gravity, that is, by the parameter Ω2R/g, where Ω is the spin speed, R is the tank radius, and g is the gravitational acceleration.

Introduction

Generally, the liquid hydrodynamic pressure in rigid containers has two distinct components. One component is directly proportional to the acceleration of the tank and is caused by part of the fluid moving in unison with the tank. The second, known as “convective” pressure, experiences sloshing at the free surface. A realistic representation of the liquid dynamics inside closed containers can be approximated by an equivalent mechanical system. The equivalence is taken in the sense of equal resulting forces and moments acting on the tank wall. By properly accounting for the equivalent mechanical system representation of sloshing, the problem of overall dynamic system behavior can be formulated more simply. For linear planar liquid motion, one can develop equivalent mechanical models in the form of a series of mass-spring dashpot systems or a set of simple pendulums. For nonlinear sloshing phenomena, other equivalent models such as spherical or compound pendulum may be developed to emulate rotational and chaotic sloshing.

Graham (1951) developed an equivalent pendulum to represent the free-surface oscillations of a liquid in a stationary tank. Graham and Rodriguez (1952) introduced another model consisting of a sloshing point mass attached with springs to the tank wall at a specified depth and a fixed rigid mass. Pinson (1964) determined spring constants for liquid propellant in ellipsoidal tanks. Models in the form of mass-spring dashpot systems or a set of simple pendulums were considered by Ewart (1956), Bauer (1960a, 1961c, 1962b), Armstrong and Kachigan (1961), Abramson and Ransleben (1961c), Mooney, et al. (1964b).

When I was a very young boy I was enchanted by airplanes. The very idea that such a machine, with no apparent motions of its own – except, of course, for that tiny rotating thing at the front – could fly through the air was amazing. I could see that birds and insects could all fly with great dexterity, but that was because they could flap their wings, and thus support their weight as well as maneuver. And fish could even “fly” through water by motions of their body. How exciting it was then, to begin to learn something about how objects interact with the fluids surrounding them, and the useful consequences of those flows. That ultimately led, of course, to the broad study of fluid dynamics, with all of its wonderful manifestations.

There is hardly a single aspect of our daily lives, and indeed even of the entire universe in which we live, that is not in some way governed or described by fluid dynamics – from the locomotion of marine animals to the birth and death of distant galaxies. As a major field of technical and scientific knowledge, there are vast bodies of literature devoted to almost every facet of fluid behavior: laminar and turbulent flows, discontinuous (separated) flows, vortex flows, internal waves, free surface waves, compressible fluids and shock waves, multi-phase flows, and many, many others. With such a countless array of fluid phenomenon before us, what then leads to the focus of the present work?

This book is an extended version of a course on continuum mechanics taught by the authors to junior graduate students in mathematics. Besides a thorough description of the fundamental parts of continuum mechanics, it contains ramifications in a number of adjacent subjects such as magnetohydrodynamics, combustion, geophysical fluid dynamics, and linear and nonlinear waves. As is, the book should appeal to a broad audience: mathematicians (students and researchers) interested in an introduction to these subjects, engineers, and scientists.

This book can be described as an “interfacial” book: interfaces between mathematics and a number of important areas of sciences. It can also be described by what it is not: it is not a book of mathematics: the mathematical language is simple, only the basic tools of calculus and linear algebra are needed. This book is not a treatise of continuum mechanics: although it contains a thorough but concise description of many subjects, it leaves aside many developments which are fundamental but not needed in practical applications and utilizations of mechanics, e.g., the intrinsic – frame invariance – character of certain quantities or the coherence of certain definitions. The reader interested by these issues is referred to the many excellent mechanics books which are available, such as those quoted in the list of references to Part I. Finally, by its size limitations, this book cannot be encyclopedic, and many choices have been made for the content; a number of subjects introduced in this book can be developed themselves into a full book.

Introduction

The dynamic analysis of a cylindrical shell experiencing elastic deformation that is comparable to its wall thickness cannot be described within the framework of the linear theory. The same is applied if the liquid free-surface amplitude is relatively large. In both cases, nonlinear analysis should be carried out. The presence of nonlinearities may result in nonlinear resonance conditions that cause complex response characteristics. One of the main difficulties in nonlinear problems of shell–liquid systems is that the boundary conditions are essentially nonlinear. This is in addition to the fact that the strain state of an elastic shell and the shape of the liquid free surface are not known a priori. The treatment of the nonlinear interaction of a liquid–shell system is a nonclassical boundary-value problem and relies on mechanics of deformable solids, fluid dynamics, and nonlinear mechanics.

With reference to nonlinear vibrations of cylindrical shells in vacuo, the literature is very rich and reports some controversies regarding the influence of nonlinearities on the shell dynamic behavior. The main results have been reviewed by Vol'mir (1972, 1979), Leissa (1973), Evensen (1974), Kubenko, et al. (1984), Amiro and Prokopenko (1997), and Amabili, et al. (1998b). Some attempts have been made to reconcile the reported discrepancies (see, e.g., Dowell, 1998, Evensen, 1999, and Amabili, et al. 1999c). It is believed that Reissner (1955) made the first attempt to study the influence of large-amplitude vibration for simply supported shells.

The purpose of this chapter is to introduce another equation describing nonlinear wave phenomena: the nonlinear Schrödinger equation (NLS), which should not to be confused with the linear Schrödinger equation from quantum mechanics (see below).

As indicated in Chapter 18, this equation, like the KdV equation, has been discovered rather recently. The two equations appear in, and are used for, wave phenomena of various types. In particular, the NLS equation, like the KdV equation, can describe water-wave phenomena, and it can also be deduced from the Euler equation of perfect fluids under appropriate hypotheses.

However, the NLS equation also describes phenomena that are very important nowadays: the propagation of waves in wave guides in relation to the design of optical long-distance communications lines and all-optical signalprocessing devices for reliable and high-bit-rate transmission of information.

Owing to the importance of the subject, and to diversify the mathematical techniques developed in this book, we will in this chapter derive the NLS equation from the Maxwell equations in the context of wave guides rather than deduce them from the Euler equations in the context of fluid mechanics.

We start in Section 20.1 by recalling the Maxwell equations, and we introduce a new phenomenon that is essential for optic fibers, namely polarization, which corresponds to, and describes, the electromagnetically anisotropic behavior of the medium.

Sloshing means any motion of the free liquid surface inside its container. It is caused by any disturbance to partially filled liquid containers. Depending on the type of disturbance and container shape, the free liquid surface can experience different types of motion including simple planar, nonplanar, rotational, irregular beating, symmetric, asymmetric, quasi-periodic and chaotic. When interacting with its elastic container, or its support structure, the free liquid surface can exhibit fascinating types of motion in the form of energy exchange between interacting modes. Modulated free surface occurs when the free-liquid-surface motion interacts with the elastic support structural dynamics in the neighborhood of internal resonance conditions. Under low gravity field, the surface tension is dominant and the liquid may be oriented randomly within the tank depending essentially upon the wetting characteristics of the tank wall.

The basic problem of liquid sloshing involves the estimation of hydrodynamic pressure distribution, forces, moments and natural frequencies of the free-liquid surface. These parameters have a direct effect on the dynamic stability and performance of moving containers.

Generally, the hydrodynamic pressure of liquids in moving rigid containers has two distinct components. One component is directly proportional to the acceleration of the tank. This component is caused by the part of the fluid moving with the same tank velocity. The second is known as “convective” pressure and represents the free-surface-liquid motion. Mechanical models such as mass-spring-dashpot or pendulum systems are usually used to model the sloshing part.

Introduction

The problem of dynamic interaction of liquid sloshing with elastic structures may fall under one of the following categories:

1. Interaction of liquid sloshing dynamics with the container elastic modes in breathing and bending. This type is addressed in this chapter and Chapter 9.

2. Interaction of liquid sloshing dynamics with the supporting elastic structure. This type is treated in Chapter 10.

3. Liquid interaction with immersed elastic structures. This class will not be addressed in this book and the reader may consult Chen (1987), Paidoussis (1998) and Dzyuba and Kubenko (2002).

This chapter presents the linear problem of liquid interaction with its elastic container. Two limiting cases may occur where interaction disappears. The first case deals with the excitation of liquid surface modes where significant elastic modes of the container are not participating. In this case, the analysis of liquid dynamics in a rigid container will provide a satisfactory description of the overall behavior. The second case deals with the excitation of the container elastic modes where significant liquid motion does not occur. In this case, the presence of liquid will contribute to the mass distributed to the tank walls, and the analysis can be carried out without considering any interaction with liquid sloshing dynamics.

The first step in studying the interaction of liquid dynamics with elastic tank dynamics is to consider the linear eigenvalue problem and response to external excitations. The coupling may take place between the liquid-free-surface dynamics and with either the tank bending oscillations or breathing modes (or shell modes).

Deformations

The purpose of mechanics is to study and describe the motion of material systems. The language of mechanics is very similar to that of set theory in mathematics: we are interested in material bodies or systems, which are made of material points or matter particles. A material system fills some part (a subset) of the ambient space (ℝ3), and the position of a material point is given by a point in ℝ3; a part of a material system is called a subsystem.

We will almost exclusively consider material bodies that fill a domain (i.e., a connected open set) of the space. We will not study the mechanically important cases of thin bodies that can be modeled as a surface (plates, shells) or as a line (beams, cables). The modeling of the motion of such systems necessitates hypotheses that are very similar to the ones we will present in this book, but we will not consider these cases here.

A material system fills a domain Ω0 in ℝ3 at a given time t0. After deformation (think, for example, of a fluid or a tennis ball), the system fills a domain Ω in ℝ3. A material point, whose initial position is given by the point a ∈ Ω0, will be, after transformation, at the point x ∈ Ω.